where we see why N[] might have a spot of trouble with evaluating the exact expression produced by Hypergeometric2F1[]: $\sum\limits_{k=1}^{m-2}\frac1{k 2^k}\approx \log\,2$, with the difference getting smaller as $m\to\infty$, and we thus see a fair amount of catastrophic cancellation during numerical evaluation. In particular, for $m=200$, $\sum\limits_{k=1}^{m-2}\frac1{k 2^k}$ and $\log\,2$ agree to $61$ (!) decimal places.

Fortunately for us, Hypergeometric2F1[] can cope nicely with inexact arguments:

Hypergeometric2F1[1, 1, N[200, 20], -1]
0.9950490265763910737

In short: just supply inexact numbers to Hypergeometric2F1[] at the outset.

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